$Lambda$-adic Families of Jacobi Forms

We show that Hidas families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-ad

A mu-ordinary Hasse invariant

We construct a generalization of the Hasse invariant for certain unitary Shimura varieties of PEL type whose vanishing locus is the complement of the so-called mu-ordinary locus. We show that the mu-ordinary locus of those varieties is affine. As an application, we strengthen a special case of a theorem of one of us (W.G.) on the association of Galois representations to automorphic representations of unitary groups whose archimedean component is a holomorphic limit of discrete series.

Cristaux et immeubles

Let G be a connected reductive group defined over Q_p. The set of crystals contained in a given G-isocrystal is viewed from a Bruhat-Tits building-theoretic vantage point as a kind of tubular neighborhood of a skeleton characterized by a minimality property arising from metric space theory.

Purity of level m stratifications

Let $k$ be a field of characteristic $p>0$. Let $D_m$ be a $BT_m$ over $k$ (i.e., an $m$-truncated Barsotti--Tate group over $k$). Let $S$ be abreak $k$-scheme and let $X$ be a $BT_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $pge 5$, we show that $S_{D_m}(X)$ is pure in $S$ i.e., the immersion $S_{D_m}(X) hookrightarrow S$ is affine. For $pin{2,3}$, we prove purity if $D_m$ satisfies a certain property depending only on its $p$-torsion $D_m[p]$. For $pge 5$, we apply the developed techniques to show that all level $m$ stratifications associated to Shimura varieties of Hodge type are pure.

Stratifications of Newton polygon strata and Traversos conjectures for p-divisible groups

The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D.